Am trying to figure out a proof that if $G$ is a finite $p$-group, then $G$ is nilpotent ... Here, nilpotent means that a group $G$ is nilpotent if there exists a series $1 = H_0 \leq H_1 \leq ... \leq H_n=G$, such that (1) $H_i$ is normal in $G$ for each $i=0...n$, and (2) $H_{i-1}/H_i \leq Z(G/H_{i-1})$, where $Z(G/H_{i-1})$ is the core of $G/H_{i-1}$.
So suppose that $G$ is a finite $p$-group. Let $Z(G)$ be the core of $G$. It can be shown that $Z(G) \neq 1$. Given that the quotient group $G/Z(G)$ is nilpotent, we have that $G$ is nilpotent as well ...
Am trying to figure out the last statement, that is, if $G/Z(G)$ is nilpotent, how does it follow that $G$ is nilpotent as well ?